October 16, 2013
Last time Arithmetic Growth Simple Interest Geometric Growth Compound Interest A limit to Compounding
Problems Question: I put $1,000 dollars in a savings account with 2% nominal interest per year. How much money will I have after 10 years? with Simple Interest? Compounded annually? Compounded quarterly? Compounded daily? Compounded continuously? Question: I had a CD with National City Bank through 2010 that paid 4.69% interest compounded daily. What was the APY fro this rate? Question: A Paper Series EE Savings Bond is sold at half face value, to the full face value by 20 years from the issue date. What is the minimum APY for such a bond
Question 1 Answer I put $1,000 dollars in a savings account with 2% nominal interest per year. How much money will I have after 10 years? with Simple Interest? Simple Interest For a principal P and an annual rate of interest r, the interest earned in t years is I = Prt and the total amount A accumulated in the account is Answer: A = P(1 + rt) A = 1000(1 + 0.02(10)) = 1000(1.2) = 1200
Question 1 Answer Compound Interest Formula for an Annual Rate An initial principal P in an account that pays interest at a a nominal annual rate r compounded m times per year, grows after t years to ( A = P 1 + r ) mt m Answer: Compounded annually A = 1000(1 + 0.02 1 )10 = 1000(1.02) 10 = 1218.99 Answer: Compounded quarterly A = 1000(1 + 0.02 4 )10(4) = 1000(1.005) 40 = 1220.79 Answer: Compounded daily A = 1000(1 + 0.02 365 )3650 = 1221.40
Question 1 Answer I put $1,000 dollars in a savings account with 2% nominal interest per year. How much money will I have after 10 years? Compounded continuously? Continuous Interest Formula For a principal P deposited in an account at a nominal annual rate r, compounded continuously, the balance after t years is A = Pe rt Answer: Compounded continuously? A = 1000e 0.02(10) = 1000e 0.2 = 1221.4027
Question 2 Answer Question: I had a CD with National City Bank through 2010 that paid 4.69% interest compounded daily. What was the APY fro this rate? Formula for APY APY = (1 + r m )m 1 where APY = annual percentage yield (effective annual rate) r = nominal interest rate m = number of compounding periods per year Answer: APY = (1 + 0.0469 365 )365 1 =.04801, 4.8%
Question 3 Answer Question: A Paper Series EE Savings Bond is sold at half face value, to the full face value by 20 years from the issue date. What is the minimum APY for such a bond? Solve: P(1 + APY ) 20 = 2P (1 + APY ) 20 = 2 1 + APY = 2 1/20 APY = 2 1/20 1 APY = 0.0352649
Problem If I put a $1,000 a year into a savings account with APY 10%. How much money will I have at the end of 5 years? Money is first deposited Jan 1, 2001 and interest is first calculated Dec 31 2001. How money will there be in the savings account on Jan 2, 2006? Answer: A = 1000 + 1000(1 +.10) + 1000(1 +.10) 2 + 1000(1 +.10) 3 + 1000(1 +.10) 4 + 1000(1 +.10) 5 A = 1000(1 + 1.1 + 1.1 2 + 1.1 3 + 1.1 4 + 1.1 5 A = 6998.95
This Time A Model for Saving Present Value and Inflation
Geometric Series Formula for Geometric Series S = a 0 + a 0 x + a 0 x 2 + a 0 x 3 + + a 0 x n 1 [ = a 0 1 + x + x 2 + x 3 + + x n] [ x n ] 1 S = a 0, x 1 x 1
Savings Formula [ (1 + i) n ] [ 1 (1 + r ] m A = d = d 1 r i m where A = amount accumulated d = regular deposit of payment at the end of each period n = mt number of periods r= nominal annual interest rate m = number of compounding periods per year t= number of years i= r/m periodic rate, the interest rate per compounding period
Payment Formula Question: What if I want 5,000 dollars in savings account with APY 10% at the end of five years, how much money will I need to deposit at the end of every year? Payment Formula [ ] i d = A (1 + i) n 1 Answer: [ = A r/m (1 + r m )mt 1.1 50000( (1.1) 5 1 ) = 818.99 ]
Savings Plans Sinking Fund A sinking fund is a savings plan to accumulate a fixed sum by a particular date, usually through equal periodic deposits. Annuity An annuity is a specified number of equal periodic payment.
Present Value and Inflation Present Value The present value of an amount to be paid or received at a specified time in the future is what that future payment would be worth today, as determined from a given interest rate and compounding period. P = A (1 + i) n = A (1 + r/m) mt Question: What is the present value of $20,000, 5 years from now, at an APY of 5%?
Present Value and Inflation Present Value The present value of an amount to be paid or received at a specified time in the future is what that future payment would be worth today, as determined from a given interest rate and compounding period. P = A (1 + i) n = A (1 + r/m) mt Question: What is the present value of $20,000, 5 years from now, at an APY of 5%? Answer: P = 20000 (1 + 0.05) 5 = 15670.50
Inflation Inflation Inflation is a rise in prices from a set base year. Annual Rate of Inflation The annual rate of inflation, a (= 100a%), is the additional proportionate cost of goods one year later. Goods that cost $ i in the base year will then cost $ (1+a). Relative Purchasing Power of a Dollar a Year from Now with Inflation Rate a $1 $a = $1 1 + a 1 + a
Example Suppose that there is constant 3% annual inflation from mid-2009 through mid-2013. What will be the projected price in mid-2013 of an item that cost $100 in mid-2009?
Example Suppose that there is constant 3% annual inflation from mid-2009 through mid-2013. What will be the projected price in mid-2013 of an item that cost $100 in mid-2009? Answer: A = 100(1 + 0.03) 4 = 112.55
Depreciation Question: How much is a mid-2009 dollar worth in terms of a mid -2013 dollar, with 3% inflation? a The lost in purchasing power due to an inflation rate of a is 1+a The relative purchasing power of P dollars t years from now as Answer: A = P(1 + i) t = P [ 1 a ] t 1 + a A = (1 0.03 1.03 )4 = 0.888487
Problems Question 1: Suppose that you want to save up $2000 for a semester abroad two years from now. How much do you have to put away each month in a savings account that earns 2 % interest compounded monthly? Question 2: A colleague feels that he will need $1 million in savings to afford to retire at age 65 and still maintain his current standard of living. Younger colleague, age 30, decides to begin savings for retirement based on that advice. How much does the younger colleague need to save per month to have $ 1 million at retirement if the fund earns a steady 3% annual interest compounded monthly? Question 3: Suppose you start saving for retirement at age 45. How much do you have to save per month, with a steady return of 6% compounded monthly, to accumulate $250,000 by age 65?
Problems Question 4: What is the present value of $10,000, 4 years from now, at an APY of 5%? Question 5: What is the present value of $15,000, 10 years from now, at an APY of 3%? Question 6: Suppose that inflation proceeds at a constant rate of 2% per year from mid- 2012 through mid 2015. a) Find the cost in mid-2015 of a basket of goods that cost $1 in mid-2012. b) What will be the value of a dollar in mid-2015 in constant mid-2012 dollars?
Next Time Real rate of Growth The real annual rate of growth of an investment at annual interest rate r with annual inflation rate a is g = r a 1 + a